Stefan A. Funken

Remarks around 50 lines of Matlab: short finite element implementation

A short Matlab implementation for P1-x1 finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documentation, any adaptation from simple model examples to more complex problems can easily be performed. Numerical examples prove the flexibility of the Matlab tool.

Matlab implementation of the finite element method in elasticity

Abstract A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.

Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second-order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, the reliability of any averaging estimator is shown for low order finite element methods in elasticity. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides and independent of the structure of a shape-regular mesh.

Fully Reliable Localized Error Control in the FEM

If the first task in numerical analysis is the calculation of an approximate solution, the second is to provide a guaranteed error bound and is often of equal importance. The standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some saturation property. For coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds. The aim of this paper is to provide reliable computable error bounds which are efficient and complete in the sense that constants are estimated as well. The main argument is a localization via a partition of unity which leads to problems on small domains. Two fully reliable estimates are established: The sharper one solves an analytical interface problem with residuals following Babuska and Rheinboldt [SIAM J. Numer. Anal., 15 (1978), pp. 736--754]. The second estimate is a modification of the standard residual-based a posteriori estimate with explicit constants from local analytical eigenvalue problems. For some class of triangulations we show that the efficiency constant is smaller than 2.5. According to our numerical experience, the overestimation of our computable estimates proved to be reasonably small, with an overestimation by a factor between 2.5 and 4 only.

A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems

Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and non-conforming finite element methods. A refined residual-based error estimate generalises the works of Verfurth; Dari, Duran and Padra; Bao and Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain nonNewtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive mesh-refining algorithms.

Averaging technique for FE – a posteriori error control in elasticity. Part II: λ-independent estimates

In the second part of our investigation on a posteriori error estimates and a posteriori error control in finite element analysis in elasticity, we focus on robust a posteriori error bounds. First we establish a residual-based a posteriori error estimate which is reliable and efficient up to higher-order terms and λ-independent multiplicative constants; the Lame constant λ steers the incompressibility. Second we show the robust efficiency and reliability of averaging techniques in certain norms. Numerical evidence supports that the reliability of depends on the smoothness of given right-hand sides and is independent of the structure of a shape-regular mesh.

Locking-free adaptive mixed finite element methods in linear elasticity

Mixed finite element methods such as PEERS or the BDMS methods are designed to avoid locking for nearly incompressible materials in plane elasticity. In this paper, we establish a robust adaptive mesh-refining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate. Numerical evidence is provided for the λ-independence of the constants in the a posteriori error bounds and for the efficiency of the adaptive mesh-refining algorithm proposed.

The BPX preconditioner for the single layer potential operator

In this paper we discuss the BPX preconditioner for the single layer potential operator. We find that the exterm eigenvalues of the preconditioner applied to the single layer petntial operator are bounded independent of the number of unknowns. A description of an efficient implemetation of the BPX algorithm is given.

A posteriori error estimates for hp--boundary element methods

This paper presents a posterior error estimates for the hp-version of the boundary element method. We dicuss two first kund integral operator equations, namely Symms integral equation and the equation with a hypersingular operator. The computable upper error bounds indicate an algorithm for the automatic hp-adaptive mesh-refnement. The efficiency of this method is shown by numerical experiments yielding almost optimal convergence even in the presence of corner singularities.

Averaging technique for a posteriori error control in elasticity. Part III: Locking-free nonconforming FEM

Abstract In the third part of our investigations on averaging techniques for a posteriori error control in elasticity we focus on nonconforming finite elements in two dimensions. Kouhia and Stenberg [Comput. Methods Appl. Mech. Engrg. 124 (1995) 195] established robust a priori error estimates for a Galerkin-discretisation where the first component of the discrete displacement function is discretised with conforming and the second with nonconforming P1 finite elements. Here we study robust, i.e., λ-independent reliability and efficiency estimates for averaging error estimators. Numerical evidence supports that the reliability depends on the smoothness of given right-hand sides and independent of the structure of a shape-regular mesh.